universe Article An Alternative to Dark Matter and Dark Energy: Scale-Dependent Gravity in Superfluid Vacuum Theory Konstantin G. Zloshchastiev Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban 4000, South Africa; [email protected] Received: 29 August 2020; Accepted: 10 October 2020; Published: 15 October 2020 Abstract: We derive an effective gravitational potential, induced by the quantum wavefunction of a physical vacuum of a self-gravitating configuration, while the vacuum itself is viewed as the superfluid described by the logarithmic quantum wave equation. We determine that gravity has a multiple-scale pattern, to such an extent that one can distinguish sub-Newtonian, Newtonian, galactic, extragalactic and cosmological terms. The last of these dominates at the largest length scale of the model, where superfluid vacuum induces an asymptotically Friedmann–Lemaître–Robertson–Walker-type spacetime, which provides an explanation for the accelerating expansion of the Universe. The model describes different types of expansion mechanisms, which could explain the discrepancy between measurements of the Hubble constant using different methods. On a galactic scale, our model explains the non-Keplerian behaviour of galactic rotation curves, and also why their profiles can vary depending on the galaxy. It also makes a number of predictions about the behaviour of gravity at larger galactic and extragalactic scales. We demonstrate how the behaviour of rotation curves varies with distance from a gravitating center, growing from an inner galactic scale towards a metagalactic scale: A squared orbital velocity’s profile crosses over from Keplerian to flat, and then to non-flat. The asymptotic non-flat regime is thus expected to be seen in the outer regions of large spiral galaxies. Keywords: quantum gravity; cosmology; superfluid vacuum; emergent spacetime; dark matter; galactic rotation curve; quantum Bose liquid; logarithmic fluid; logarithmic wave equation 1. Introduction Astronomical observations over many length scales support the existence of a number of novel phenomena, which are usually attributed to dark matter (DM) and dark energy (DE). Dark matter was introduced to explain a range of observed phenomena at a galactic scale, such as flat rotation curves, while dark energy is expected to account for cosmological-scale dynamics, such as the accelerating expansion of the Universe. For instance, the LCDM model, which is currently the most popular approach used in cosmology and galaxy-scale astrophysics, makes use of both DE and cold DM concepts [1]. In spite of being a generally successful framework purporting to explain the large-scale structure of the Universe, it currently faces certain challenges [2,3]. There is also growing consensus that a convincing theory of DM- and DE-attributed phenomena cannot be a stand-alone model; but should, instead, be a part of a fundamental theory involving all known interactions. In turn, we contend that formulating this fundamental theory will be impossible without a clear understanding of the dynamical structure of the physical vacuum, which underlies all interactions that we know of. Moreover, this theory must operate at a quantum level, which necessitates us rethinking of the concept of gravity using basic notions of quantum mechanics. Universe 2020, 6, 180; doi:10.3390/universe6100180 www.mdpi.com/journal/universe Universe 2020, 6, 180 2 of 25 One of the promising candidates for a theory of physical vacuum is superfluid vacuum theory (SVT), a post-relativistic approach to high-energy physics and gravity. Historically, it evolved from Dirac’s idea of viewing the physical vacuum as a nontrivial quantum object, whose phase and derived velocity are non-observable in a quantum-mechanical sense [4]. The term ‘post-relativistic’, in this context, means that SVT can generally be a non-relativistic theory; which nevertheless contains relativity as a special case, or limit, with respect to some dynamical value such as momentum (akin to general relativity being a superset of the Newton’s theory of gravity). Therefore, underlying three-dimensional space would not be physically observable until an observer goes beyond the above-mentioned limit, as will be discussed in more detail later in this article. The dynamics and structure of superfluid vacuum are being studied, using various approaches which agree upon the main paradigm (physical vacuum being a background quantum liquid of a certain kind, and elementary particles being excitations thereof), but differ in their physical details, such as an underlying model of the liquid [5–7]. It is important to work with a precise definition of superfluid, to ensure that we avoid the most common misconceptions which otherwise might arise when one attempts to apply superfluid models to astrophysics and cosmology, some details can be found in AppendixA. In fact, some superfluid-like models of dark matter based on classical perfect fluids, scalar field theories or scalar-tensor gravities, turned out to be vulnerable to experimental verification [8]. Moreover, superfluids are often confused not only with perfect fluids, but also with the concomitant phenomenon of Bose-Einstein condensates (BEC), which is another kind of quantum matter occurring in low-temperature condensed matter [9]. However, even though BEC’s do share certain features with superfluids, this does not imply that they are superfluidic in general. In particular, quantum excitations in laboratory superfluids that we know of have dispersion relations of a distinctive shape called the Landau “roton” spectrum. Such a shape of the spectral curve is crucial, as it ensures the suppression of dissipative fluctuations at a quantum level [10,11], which results in inviscid flow [12,13]. If plotted as an excitation energy versus momentum, the curve starts from the origin, climbs up to a local maximum (called the maxon peak), then slightly descends to a local nontrivial minimum (called the “roton” energy gap); then grows again, this time all the way up, to the boundary of the theory’s applicability range. In fact it is not the roton energy gap alone, but the energy barrier formed by the maxon peak and roton minimum in momentum space, which ensures the above-mentioned suppression of quantum fluctuations in quantum liquid and, ultimately, causes its flow to become inviscid. In other words, it is the global characteristics of the dispersion curve, not just the existence of a nontrivial local minimum and related energy gap, which is important for superfluidity to occur. Obviously, these are non-trivial properties, which cannot possibly occur in all quantum liquids and condensates. Further details and aspects are discussed in AppendixA. This paper is organized as follows. Theory of physical vacuum based on the logarithmic superfluid model is outlined in Section2, where we also demonstrate how four-dimensional spacetime can emerge from the three-dimensional dynamics of quantum liquid. In Section3, we derive the gravitational potential, induced by the logarithmic superfluid vacuum in a given state, using certain simplifying assumptions. Thereafter, in Section4, we give a brief physical interpretation of different parts of the derived gravitational potential and estimate their characteristic length scales. In Section5, profiles of induced matter density are derived and discussed for the case of spherical symmetry. Galactic scale phenomena are discussed in Section6, where the phenomenon of galactic rotation curves is explained without introducing any exotic matter ad hoc. In Section7, we discuss the various mechanisms of the accelerating expansion of the Universe, as well as the cosmological singularity, “vacuum catastrophe” and cosmological coincidence problems. Conclusions are drawn in Section8. Universe 2020, 6, 180 3 of 25 2. Logarithmic Superfluid Vacuum Superfluid vacuum theory assumes that the physical vacuum is described, when disregarding quantum fluctuations, by the fluid condensate wavefunction Y(r, t), which is a three-dimensional Euclidean scalar. The state itself is described by a ray in the corresponding Hilbert space, therefore this wavefunction obeys a normalization condition Z hYjYi = r dV = M, (1) V where M and V are the total mass and volume of the fluid, respectively, and r = jYj2 is the fluid mass density. The wavefunction’s dynamics is governed by an equation of a U(1)-symmetric Schrödinger form: " # h¯ 2 −ih¯ ¶ − r2 + V (r, t) + F(jYj2) Y = 0, (2) t 2m ext where m is the constituent particles’ mass, Vext(r, t) is an external or trapping potential and F(r) is a duly chosen function, which effectively takes into account many-body effects inside the fluid. This wave equation can be formally derived as a minimizing condition of an action functional with the following Lagrangian: ih¯ h¯ 2 L = (Y¶ Y∗ − Y∗¶ Y) + jrYj2 + V (r, t) jYj2 + V(jYj2), (3) 2 t t 2m ext where V(r) equals to a primitive of F(r) up to an additive constant: F(r) = V0(r); throughout the paper the prime denotes a derivative with respect to the function’s argument. In this picture, massless excitations, such as photons, are analogous to acoustic waves propagating p 0 with velocity cs ∝ jp (r)j, where fluid pressure p = p(r) is determined via the equation of state. For the system (2), both the equation of state and speed of sound can be derived using the fluid-Schrödinger analogy, which was established for a special case in [14], and generalized for an arbitrary F(r) in works [7,15]. In a leading-order approximation with respect to the Planck constant, we obtain 1 Z 1 p = − rF0(r) dr, c2 = rjF0(r)j, (4) m s m while higher-order corrections would induce Korteweg-type effects, thus significantly complicating the subject matter [15]. Furthermore, it is natural to require that superfluid vacuum theory must recover Einstein’s theory of relativity at a certain limit. One can show that at a limit of low momenta of quantum excitations, often called a “phononic” limit by analogy with laboratory quantum liquids, Lorentz symmetry does emerge.